Showing papers on "Convex optimization published in 1994"
TL;DR: In this paper, the continuous and discrete-time H∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI), and two interesting new features emerge through this approach: solvability conditions valid for both regular and singular problems, and an LMI-based parametrization of all H ∞-suboptimal controllers, including reduced-order controllers.
Abstract: The continuous- and discrete-time H∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI). Two interesting new features emerge through this approach: solvability conditions valid for both regular and singular problems, and an LMI-based parametrization of all H∞-suboptimal controllers, including reduced-order controllers.
The solvability conditions involve Riccati inequalities rather than the usual indefinite Riccati equations. Alternatively, these conditions can be expressed as a system of three LMIs. Efficient convex optimization techniques are available to solve this system. Moreover, its solutions parametrize the set of H∞ controllers and bear important connections with the controller order and the closed-loop Lyapunov functions.
Thanks to such connections, the LMI-based characterization of H∞ controllers opens new perspectives for the refinement of H∞ design. Applications to cancellation-free design and controller order reduction are discussed and illustrated by examples.
3,200 citations
TL;DR: The existence conditions are equivalent to Scherer's results, but with a more elementary derivation, and the set of all H∞ controllers explicitly parametrized in the state space using the positive definite solutions to the LMIs is provided.
1,253 citations
TL;DR: Using an extension of Pierra's product space formalism, it is shown here that a multiprojection algorithm converges and is fully simultaneous, i.e., it uses in each iterative stepall sets of the convex feasibility problem.
Abstract: Generalized distances give rise to generalized projections into convex sets. An important question is whether or not one can use within the same projection algorithm different types of such generalized projections. This question has practical consequences in the area of signal detection and image recovery in situations that can be formulated mathematically as a convex feasibility problem. Using an extension of Pierra's product space formalism, we show here that a multiprojection algorithm converges. Our algorithm is fully simultaneous, i.e., it uses in each iterative stepall sets of the convex feasibility problem. Different multiprojection algorithms can be derived from our algorithmic scheme by a judicious choice of the Bregman functions which govern the process. As a by-product of our investigation we also obtain blockiterative schemes for certain kinds of linearly constraned optimization problems.
1,085 citations
1,059 citations
Book•
[...]
01 Nov 1994
TL;DR: The first two chapters of the book as mentioned in this paper are devoted to convexity in the classical sense, for functions of one and several real variables respectively, giving a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets and sets which are convex for supports or singular supports with respect to a differential operator.
Abstract: The first two chapters of the book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets and sets which are convex for supports or singular supports with respect to a differential operator. In addition the convexity conditions which are relevant for local or global existence of holomorphic solutions of holomorphic differential equations are discussed, leading up to Trepreau's theorem on sufficiency of condition (psi) for microlocal solvability in the analytic category.
734 citations
TL;DR: In this article, a parameter-dependent control problem for linear parametrically varying (LPV) systems is presented, where sufficient conditions are given that guarantee an LPV system is exponentially stable and achieves an induced L 2 -norm performance objective from the disturbance to error signals.
644 citations
29 Jun 1994
TL;DR: In this article, the authors address the robustness issue in MPC by directly incorporating the description of plant uncertainty in the MPC problem formulation, where the plant uncertainty is expressed in the time domain by allowing the state-space matrices of the discrete-time plant to be arbitrarily time-varying and belonging to a polytope.
Abstract: The primary disadvantage of current design techniques for model predictive control (MPC) is their inability to explicitly deal with model uncertainty. In this paper, the authors address the robustness issue in MPC by directly incorporating the description of plant uncertainty in the MPC problem formulation. The plant uncertainty is expressed in the time-domain by allowing the state-space matrices of the discrete-time plant to be arbitrarily time-varying and belonging to a polytope. The existence of a feedback control law minimizing an upper bound on the infinite horizon objective function and satisfying the input and output constraints is reduced to a convex optimization over linear matrix inequalities (LMIs). It is shown that for the plant uncertainty described by the polytope, the feasible receding horizon state feedback control design is robustly stabilizing.
621 citations
29 Jun 1994
TL;DR: In this article, the authors give a historical view of linear matrix inequalities in control and system theory, and it appears that linear matrices have been involved in some of the major events of control theory.
Abstract: The purpose of this paper is to give a historical view of linear matrix inequalities in control and system theory. Not surprisingly, it appears that LMIs have been,involved in some of the major events of control theory. With the advent of powerful convex optimization techniques, LMIs are now about to become an important practical tool for future control applications.
484 citations
TL;DR: This paper presents a decomposition method for solving convex minimization problems that preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation.
Abstract: This paper presents a decomposition method for solving convex minimization problems. At each iteration, the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables. We derive this algorithm from Rockafellar's proximal method of multipliers, which involves an augmented Lagrangian with an additional quadratic proximal term. The algorithm preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation. We allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem's data, the method is globally convergent and at a linear rate. The method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets.
420 citations
TL;DR: This work uses time-domain input-output data to validate uncertainty models and develops algorithms that are computationally tractable and reduce to (generally nondifferentiable) convex feasibility problems or to linear programming problems.
Abstract: In this paper we offer a novel approach to control-oriented model validation problems. The problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input-output data. Our approach directly uses time-domain input-output data to validate uncertainty models. The algorithms we develop are computationally tractable and reduce to (generally nondifferentiable) convex feasibility problems or to linear programming problems. In special cases, we give analytical solutions to these problems. >
372 citations
TL;DR: In this paper, a Riccati equation approach is proposed to solve the problem of Kalman filter design for uncertain systems and a suboptimal covariance upper bound can be computed by a convex optimization.
28 Sep 1994
TL;DR: A new application of the abstract interpretation by means of convex polyhedra to a class of hybrid systems, i.e., systems involving both discrete and continuous variables, is presented.
Abstract: We present a new application of the abstract interpretation by means of convex polyhedra, to a class of hybrid systems, i.e., systems involving both discrete and continuous variables. The result is an efficient automatic tool for approximate, but conservative, verification of reachability properties of these systems.
Book•
31 Mar 1994
TL;DR: In this article, the authors introduce the logarithmic barrier method (LBP) and the center method (CPM) for reducing the complexity of LP and CPM, respectively.
Abstract: Glossary of Symbols and Notations. 1. Introduction of IPMs. 2. The logarithmic barrier method. 3. The center method. 4. Reducing the complexity for LP. 5. Discussion of other IPMs. 6. Summary, conclusions and recommendations. Appendices: A. Self-concordance proofs. B. General technical lemmas. Bibliography. Index.
TL;DR: Using a powerful technique involving quadratic or higher order potential functions, methods for deriving polyhedral and nonlinear sets that contain the set of achievable response times under stable and preemptive scheduling policies are proposed and found that the first order approximation of the method is at least as good as simulation-based existing methods.
Abstract: We consider open and closed multiclass queueing networks, with Poisson arrivals (for open networks), exponentially distributed class dependent service times and class dependent deterministic or probabilistic routing. The performance objective is to minimize, over all sequencing and routing policies, a weighted sum of the expected response times of different classes. Using a powerful technique involving quadratic or higher order potential functions, we propose methods for deriving polyhedral and nonlinear sets that contain the set of achievable response times under stable and preemptive scheduling policies. By optimizing over these sets, we obtain lower bounds on achievable performance. In the special case of single station networks (multiclass queues and Klimov's model) and homogeneous multiclass networks, the polyhedron derived is exactly equal to the achievable region. Consequently, the proposed method can be viewed as the natural extension of conservation laws to multiclass queueing networks. We apply the same approach to closed networks to obtain upper bounds on the optimal throughput. We check the tightness of our bounds by simulating heuristic policies and we find that the first order approximation of our method is at least as good as simulation-based existing methods. In terms of computational complexity and in contrast to simulation-based existing methods, the calculation of our first order bounds consists of solving a linear programming problem with a number of variables and constraints that is polynomial (quadratic) in the number of classes in the network. The $i$th order approximation leads to a convex programming problem in dimension $O(R^{i+1})$, where $R$ is the number of classes in the network, and can be solved efficiently using techniques from semidefinite programming.
TL;DR: The proposed deterministic annealing neural network is shown to be capable of generating optimal solutions to convex programming problems and the conditions for asymptotic stability, solution feasibility, and solution optimality are derived.
TL;DR: A global optimization algorithm is proposed based on an efficient partitioning strategy which is guaranteed to attain ε-convergence to the global minimum potential energy configuration of a molecule through the solution of a series of nonlinear convex optimization problems.
Abstract: A global optimization algorithm is proposed for finding the global minimum potential energy conformations of small molecules. The minimization of the total potential energy is formulated on an independent set of internal coordinates involving only torsion (dihedral) angles. Analytical expressions for the Euclidean distances between non-bonded atoms, which are required for evaluating the individual pairwise potential terms, are obtained as functions of bond lengths, covalent bond angles, and torsion angles. A novel procedure for deriving convex lower bounding functions for the total potential energy function is also introduced. These underestimating functions satisfy a number of important theoretical properties. A global optimization algorithm is then proposed based on an efficient partitioning strategy which is guaranteed to attain e-convergence to the global minimum potential energy configuration of a molecule through the solution of a series of nonlinear convex optimization problems. Moreover, lower and upper bounds on the total finite number of required iterations are also provided. Finally, this global optimization approach is illustrated with a number of example problems.
29 Jun 1994
TL;DR: In this article, a polynomial-time projective algorithm for the numerical solution of linear matrix inequalities (LMI) is presented, and a complexity analysis is provided, and applications to two generic LMI problems are discussed.
Abstract: In many control problems, the design constraints have natural formulations in terms of linear matrix inequalities (LMI). When no analytical solution is available, such problems can be attacked by solving the LMIs via convex optimization techniques. This paper describes the polynomial-time projective algorithm for the numerical solution of LMIs. Simple geometrical arguments are used to clarify the strategy and convergence mechanism of the projective method. A complexity analysis is provided, and applications to two generic LMI problems are discussed.
TL;DR: Convergence results for both the alternating step and epigraphic methods are given, and their performance on random dense separable quadratic programs is compared.
Abstract: This paper describes two specializations of the alternating direction method of multipliers: the alternating step method and the epigraphic projection method. The alternating step method applies to monotropic programs, while the epigraphic method applies to general block-separable convex programs, including monotropic programs as a special case. The epigraphic method resembles an earlier parallel method due to Spingarn, but solves a larger number of simpler subproblems at each iteration. This paper gives convergence results for both the alternating step and epigraphic methods, and compares their performance on random dense separable quadratic programs.
TL;DR: An extension of the proximal method for convex programming, where the quadratic regularization kernel is substituted by a class of convex statistical distances, called φ- divergences, is studied and a convergence analysis of the resulting entropy-like proximal algorithm is presented.
Abstract: We study an extension of the proximal method for convex programming, where the quadratic regularization kernel is substituted by a class of convex statistical distances, called φ-divergences, which are typically entropy-like in form. After establishing several basic properties of these quasi-distances, we present a convergence analysis of the resulting entropy-like proximal algorithm. Applying this algorithm to the dual of a convex program, we recover a wide class of nonquadratic multiplier methods and prove their convergence.
TL;DR: This paper presents block-coordinate descent algorithms for the approximate solution of large structured convex programming problems and shows that feasibility as well as min-mix resource-sharing problems for such constraints can be solved to a relative accuracy.
Abstract: This paper presents block-coordinate descent algorithms for the approximate solution of large structured convex programming problems. The constraints of such problems consist of K disjoint convex compact sets $B^k $ called blocks, and M nonnegative-valued convex block-separable inequalities called coupling or resource constraints. The algorithms are based on an exponential potential function reduction technique. It is shown that feasibility as well as min-mix resource-sharing problems for such constraints can be solved to a relative accuracy $\varepsilon$ in $O( K\ln M ( \varepsilon^{ - 2} + \ln K ) )$ iterations, each of which solves K block problems to a comparable accuracy, either sequentially or in parallel. The same bound holds for the expected number of iterations of a randomized variant of the algorithm which uniformly selects a random block to process at each iteration. An extension to objective and constraint functions of arbitrary sign is also presented. The above results yield fast approximatio...
TL;DR: The decentralized control problem for linear dynamic systems is revisited using a parameter space approach which enables the definition of the decentralized feedbacks from the existence of non-empty parameter convex sets.
TL;DR: Conditions of existence of a homogeneous polynomial Lyapunov function of an arbitrary even degree establishing global asymptotic stability of linear system with box-bounded uncertainty are derived.
Abstract: In this note, we derive conditions of existence of a homogeneous polynomial Lyapunov function of an arbitrary even degree establishing global asymptotic stability of linear system with box-bounded uncertainty. Verification of these conditions is reduced to solving a convex minimization problem. We produce numerical examples that demonstrate significant improvement in estimates of admissible uncertainty bounds compared with estimates obtained via the most commonly used quadratic Lyapunov functions. >
TL;DR: In this article, two variations of the proximal method of multipliers, or classical augmented Lagrangian method for convex programming, are compared, and convergence is shown for a method combining the features of these two variations.
Abstract: Consider two variations of the method of multipliers, or classical augmented Lagrangian method for convex programming. The proximal method of multipliers adjoins quadratic primal proximal terms to the augmented Lagrangian, and has a stronger primal convergence theory than the standard method. On the other hand, the alternating direction method of multipliers, which uses a special kind of partial minimization of the augmented Lagrangian, is conducive to the derivation of decomposition methods finding application in parallel computing. This note shows convergence a method combining the features of these two variations. The method is closely related to some algorithms of Gols'shtein. A comparison of the methods helps illustrate the close relationship between previously separate bodies of Western and Soviet literature.
01 Jan 1994
TL;DR: The alternating direction method of multipliers decomposition algorithm for convex programming, as recently generalized by Eckstein and Bert- sekas, is considered, and some reformulations of the algorithm are given, and several alternative means for deriving them are discussed.
Abstract: We consider the alternating direction method of multipliers decomposition algorithm for convex programming, as recently generalized by Eckstein and Bert- sekas. We give some reformulations of the algorithm, and discuss several alternative means for deriving them. We then apply these reformulations to a number of optimization problems, such as the minimum convex-cost transportation and multicommodity flow. The convex transportation version is closely related to a linear-cost transportation algorithm proposed earlier by Bertsekas and Tsitsiklis. Finally, we construct a simple data-parallel implementation of the convex-cost transportation algorithm for the CM-5 family of parallel computers, and give computational results. The method appears to converge quite quickly on sparse quadratic-cost transportation problems, even if they are very large; for example, we solve problems with over a million arcs in roughly 100 iterations, which equates to about 30 seconds of run time on a system with 256 processing nodes. Substantially better timings can probably be achieved with a more careful implementation.
TL;DR: The smoothing approximation is used as the basis of an algorithm for solving problems with embedded network structures, and nonlinear minimax problems, demonstrating the efficiency of this approach.
Abstract: A quadratic smoothing approximation to nondifferentiable exact penalty functions for convex constrained optimization is proposed and its properties are established The smoothing approximation is used as the basis of an algorithm for solving problems with (i) embedded network structures, and (ii) nonlinear minimax problems Extensive numerical results with large-scale problems illustrate the efficiency of this approach
TL;DR: In this paper, a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations is presented. But the main contribution of this paper is to give a new parameterization that can be expressed in terms of quasi-convex optimization.
Abstract: This paper treats synthesis of robust controllers for linear time-invariant systems. Uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial. The main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations. With the new parameterization, certain robust performance problems can be stated in terms of quasi-convex optimization. >
TL;DR: The proposed branch and bound type algorithm attains finite e‐convergence to the global minimum through the successive refinement of converging lower and upper bounds on the solution through a novel convex lowering bounding of the total potential function and the subsequent solution of a series of nonlinear convex optimization problems.
Abstract: A deterministic global optimization algorithm is introduced for locating global minimum potential energy molecular conformations. The proposed branch and bound type algorithm attains finite e‐convergence to the global minimum through the successive refinement of converging lower and upper bounds on the solution. These bounds are obtained through a novel convex lowering bounding of the total potential function and the subsequent solution of a series of nonlinear convex optimization problems. The minimization of the total potential energy function is performed on an independent set of internal coordinates involving only dihedral angles. A number of example problems illustrate the proposed approach.
29 Jun 1994
TL;DR: In this paper, it was shown that the existence of an output feedback gain is equivalent to a positive definite matrix belonging to a convex set such that its inverse belongs to another convex setting.
Abstract: The main objective of this paper is to solve the following stabilizing output feedback control problem. Given matrices (A, B/sub 2/, C/sub 2/) with appropriate dimensions, find (if one exists), a static output feedback gain L such that the closed-loop matrix A-B/sub 2/LC/sub 2/ is asymptotically stable. Using linear matrix inequalities, it is shown that the existence of L is equivalent to the existence of a positive definite matrix belonging to a convex set such that its inverse belongs to another convex set. Conditions are provided for global convergence of the min/max algorithm which decomposes the determination of the aforementioned matrix by a sequence of convex programs. Some examples borrowed from the literature are solved hi order to illustrate the theoretical results.
TL;DR: This note proposes a new method, based on convex programming, for solving the linear quadratic problem (LQP) directly on the parameter space generated by the feedback control gain, as a minimization of a linear function over this convex set.
Abstract: This note proposes a new method, based on convex programming, for solving the linear quadratic problem (LQP) directly on the parameter space generated by the feedback control gain. All stabilizing controllers are mapped into a convex set; the problem is then formulated as a minimization of a linear function over this convex set. Its optimal solution furnishes, under certain conditions, the same feedback control gain obtained from the classical Riccati equation. Generalizations to decentralized control and output feedback control design are included. The theory is illustrated by some numerical examples. >
TL;DR: An algorithm for generalized convex multiplicative programming problems, a special class of nonconvex minimization problems in which the objective function is expressed as a sum ofp products of two convex functions, is discussed.
Abstract: This paper discusses an algorithm for generalized convex multiplicative programming problems, a special class of nonconvex minimization problems in which the objective function is expressed as a sum ofp products of two convex functions. It is shown that this problem can be reduced to a concave minimization problem with only 2p variables. An outer approximation algorithm is proposed for solving the resulting problem.